Compressed sensing is described by E. J. Candès, J. Romberg, and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information,” IEEE Trans. Information Theory, Vol. 52, No. 2, February 2006; D. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory, Vol. 52, No. 4, April 2006; and R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Proc. Mag., Vol. 118, July 2007 builds on the theoretical results developed by Candès, et. al. and Donoho who showed that a signal x of length N having a k-sparse representation of the form x=Ψα in the basis Ψ can be reconstructed from a small number M of randomly sampled linear projections α. Currently, in most existing compressed sensing techniques one does not measure or encode the k significant components of a directly. Rather, M<N (often random) projections)=y(m)=<x,φ(m)> of the signal onto a second basis φ are encoded. In matrix notation, y=Φx is measured where y is an M×1 column vector and the measurement basis matrix is M×N. Since M<N, recovery of the signal x from the measurements y is ill-posed. However, compressed sensing theory tells us that for sparse signals, when the basis Φ cannot sparsely represent the basis Ψ, i.e., when the two basis are incoherent, then it is possible to recover the large set of coefficients α and hence the signal x, provided that the number of measurements M is large enough.
There are two main drawbacks in existing compressed sensing methods, often based on random measurement bases. In many practical systems such as computed tomography, SAR, and MRI, random measurement is incompatible with the reconstruction requirements; In random unstructured sampling both computational and memory requirements are almost impossible to meet, defeating the whole purpose of compressed sensing. For instance, for N of the order of 106, measurements of M=25,000 would require more than 3 Gigabytes of memory just to store Φ and a Gigaflop to apply it. Thus the immediate challenge is whether it is possible to incorporate a priori information in order to choose the sparse set of measurements adaptively.
Recently, several methods have been proposed to find specific bases using training sets. Alternatively, a technique is also proposed to exploit causality of signals in time in order to adapt current measurements based on past ones. These methods either rely on examples or require causality, e.g., video data to achieve adaptive measurements.
Compressed sensing is generally defined as a new method to directly acquire a smaller M (M<<N) sample set signal called the compressed signal without going through the intermediate step of acquiring all N samples of the signal. As described in R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Proc. Mag., Vol. 118, July 2007 and Dharmpal Takhar et al., “A New Compressive Imaging Camera Architecture using Optical-Domain Compression,” Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 6065, 606509, 2006, they used 2-D random codes to directly implement what they say is compressed optical sensing using the DMD SLM single point detector imager design earlier demonstrated by co-inventor N. A. Riza and co-workers.
The R. G. Baraniuk system uses 1600 random measurements taken by the imager to enable compressed sensing to non-adaptively generate the compressed signal. Note that 1600 binary 2-D image masks have to be programmed onto the DMD. In effect, the sensor SLM drive hardware must use a large (e.g., 1600) random mask set, each mask pixel count is the same size as the image to be, recovered for good image reconstruction. This in-turn defeats the purpose of compressed sensing.
To solve the problems associated with the prior art systems, methods and systems of the present invention provide a novel type of compressed sensing camera for optical imaging applications. The sensing optical imager directly acts on the light to deliver the compressed signal without requiring massive data storage or complex computations.